To determine which transitions of a
He+ion can produce a spectral line with the same wavelength as a given transition in a hydrogen atom, let's examine the Rydberg formula:
For a hydrogen atom (
Z=1 ), the wavelength
λ1 for a transition from
n2 to
n1 is given by:
‌=R(‌−‌)Here,
R is the Rydberg constant, and
n1 and
n2 are the principal quantum numbers with
n2>n1.
For a
He+ion (with
Z=2 ), if the electron transitions from
n4 to
n3, the wavelength
λ2 is given by:
‌=R⋅4(‌−‌)Assuming
λ1=λ2, we have:
R(‌−‌)=R⋅4(‌−‌)By comparing both sides, we get the relationships:
n1=‌,‌‌n2=‌Both
n1,n2,n3, and
n4 must be integers, as they represent principal quantum numbers.
This condition is satisfied when
n3=2 and
n4=4, ensuring that transitions corresponding to these quantum numbers yield the same wavelength for both
He+and hydrogen.